Triple Adjointness. We have a triple adjunction witnessed by bijections of sets
\begin{align*} \textup{Hom}_{\mathcal{P}\webleft (B\webright )}\webleft (f_{*}\webleft (U\webright ),V\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,f^{-1}\webleft (V\webright )\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (f^{-1}\webleft (U\webright ),V\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,f_{!}\webleft (V\webright )\webright ), \end{align*}
natural in $U\in \mathcal{P}\webleft (A\webright )$ and $V\in \mathcal{P}\webleft (B\webright )$ and (respectively) $V\in \mathcal{P}\webleft (A\webright )$ and $U\in \mathcal{P}\webleft (B\webright )$, i.e. where:
- The following conditions are equivalent:
- We have $f_{*}\webleft (U\webright )\subset V$.
- We have $U\subset f^{-1}\webleft (V\webright )$.
- The following conditions are equivalent:
- We have $f^{-1}\webleft (U\webright )\subset V$.
- We have $U\subset f_{!}\webleft (V\webright )$.